Chapter 1.4, Problem 30E

a.

To determine

Graph the function.

Answer to Problem 30E

  

Explanation of Solution

Given information:

We consider the family of root function $f\left(x\right)=\frac{1}{x^n}$ , where $n$ is a positive integer. Graph the function $y=1/x,and,y=1/x^3$ on the same screen using the viewing rectangle $\left[-3,3\right]$ by $\left[-3,3\right]$

Calculation:

Consider the given function, $f\left(x\right)=\frac{1}{x^n}$ and sketch the graph for $y=1/x,and,y=1/x^3$ . using the viewing rectangle $\left[-3,3\right]$ by $\left[-3,3\right]$ .

  

Hence, the result shown in the graph.

b.

To determine

Graph the function.

Answer to Problem 30E

  

Explanation of Solution

Given information:

We consider the family of root function $f\left(x\right)=\frac{1}{x^n}$ , where $n$ is a positive integer. Graph the function $y=1/x^2,and,y=1/x^4$ on the same screen using the viewing rectangle $\left[-3,3\right]$ by $\left[-3,3\right]$

Calculation:

Consider the given function, $f\left(x\right)=\frac{1}{x^n}$ and sketch the graph for $y=1/x,and,y=1/x^3$ . using the viewing rectangle $\left[-3,3\right]$ by $\left[-3,3\right]$ .

  

Hence, the result shown in the graph.

c.

To determine

Graph the function.

Answer to Problem 30E

  

  

Explanation of Solution

Given information:

We consider the family of root function $f\left(x\right)=\frac{1}{x^n}$ , where $n$ is a positive integer. Graph the function $y=1/x^2,and,y=1/x^4$ , $y=1/x,and,y=1/x^3$ on the same screen using the viewing rectangle $\left[-1,3\right]$ by $\left[-1,3\right]$

Calculation:

Consider the given function, $f\left(x\right)=\frac{1}{x^n}$ and sketch the graph for $y=1/x,and,y=1/x^3$ , $y=1/x,and,y=1/x^3$ using the viewing rectangle $\left[-1,3\right]$ by $\left[-1,3\right]$ .

  

  

Hence, the result shown in the graph.

d.

To determine

Conclude these graphs.

Answer to Problem 30E

All points are satisfactorily concluded.

Explanation of Solution

Given information:

We consider the family of root function $f\left(x\right)=\frac{1}{x^n}$ , where $n$ is a positive integer. What conclusion can you make from these graphs.

Calculation:

Consider the given function, $f\left(x\right)=\frac{1}{x^n}$ and sketch the graph for $y=1/x,and,y=1/x^3$ , $y=1/x,and,y=1/x^3$ using the viewing rectangle $\left[-1,3\right]$ by $\left[-1,3\right]$ .

  

From these above plotted graphs,

1. The graphs of all functions of the form $f\left(x\right)=\frac{1}{x^n}$ is moving clockwise as $n$increases .
2. All passes through the point $\left[1,1\right]$ .
3. If $n$ is even the function plotted above $x-axis$ .
4. If $n$is odd the function is negative for negative and vice-versa.
5. As $n$ increase the graphs of $f\left(x\right)=\frac{1}{x^n}$ approaches zero faster as $x$increases

Hence, all points are satisfactory concluded.